• Open Access

Paul K. Faehrmann1,2, Jens Eisert1,3, and Richard Kueng2

• 1Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
• 2Institute for Integrated Circuits and Quantum Computing, Johannes Kepler University Linz, Linz, Austria
• 3Helmholtz-Zentrum Berlin für Materialien und Energie, Hahn-Meitner-Platz 1, 14109 Berlin, Germany

Abstract

The Hadamard test is one of the pillars on which quantum algorithm development rests and, at the same time, is naturally suited for the intermediate regime between the current era of noisy quantum devices and complete fault tolerance. Its applications use measurements of the auxiliary qubit to extract information but disregard the system register completely. Concomitantly, but independently of this development, advances in quantum learning theory have enabled the efficient representation of quantum states via classical shadows. This Letter shows that, strikingly, putting both lines of thought into a new context results in substantial improvements to the Hadamard test on a single auxiliary readout qubit, by suitably exploiting classical shadows on the remaining n-qubit work register. We argue that this combination inherits the best of both worlds and discuss statistical phase estimation as a vignette application. At the same time, the framework is more general and applicable to a wide range of other algorithms. There, we can use the Hadamard test to estimate energies on the auxiliary qubit, while classical shadows on the system register provide access to additional features such as (i) the fidelity of the initial state with certain pure quantum states, (ii) the initial state’s energy, and (iii) how pure and how close the initial state is to an eigenstate of the Hamiltonian. Finally, we also discuss how anticontrolled unitaries can further augment this framework and explain how this Letter settles the exploitation of the Hadamard test for intermediate applications.

• Quantum algorithms
• Quantum algorithms & computation

Article Text

Supplemental Material

References (57)

1. B. W. Reichardt, D. Aasen, R. Chao, A. Chernoguzov, W. van Dam, J. P. Gaebler, D. Gresh, D. Lucchetti, M. Mills, S. A. Moses, B. Neyenhuis, A. Paetznick, A. Paz, P. E. Siegfried, M. P. da Silva, K. M. Svore, Z. Wang, and M. Zanner, Demonstration of quantum computation and error correction with a tesseract code, arXiv:2409.04628.
2. D. Bluvstein, S. J. Evered, A. A. Geim, S. H. Li, H. Zhou, T. Manovitz, S. Ebadi, M. Cain, M. Kalinowski, D. Hangleiter, J. P. Bonilla Ataides, N. Maskara, I. Cong, X. Gao, P. Sales Rodriguez, T. Karolyshyn, G. Semeghini, M. J. Gullans, M. Greiner, V. Vuletić, and M. D. Lukin, Logical quantum processor based on reconfigurable atom arrays, Nature (London) 626, 58 (2024).
3. Google Quantum AI and Collaborators, Quantum error correction below the surface code threshold, Nature (London) 638, 920 (2025).
4. B. W. Reichardt et al., Logical computation demonstrated with a neutral atom quantum processor, arXiv:2411.11822.
5. H. Putterman, K. Noh, C. T. Hann et al., Hardware-efficient quantum error correction via concatenated bosonic qubits, Nature (London) 638, 927 (2025).
6. J. Preskill, Quantum computing in the NISQ era and beyond, Quantum 2, 79 (2018).
7. J. M. Arrazola, From NISQ to ISQ, (2023), https://pennylane.ai/blog/2023/06/from-nisq-to-isq/ (accessed Jan 13, 2025).
8. J. Preskill, Beyond NISQ: The megaquop machine, ACM Trans. Quantum Comput. 6, 1 (2025).
9. N. Koukoulekidis, S. Wang, T. O’Leary, D. Bultrini, L. Cincio, and P. Czarnik, A framework of partial error correction for intermediate-scale quantum computers, Report No. LA-UR-23-26230, 2023, arXiv:2306.15531.
10. A. Montanaro, Quantum algorithms: An overview, npj Quantum Inf. 2, 15023 (2016).
11. A. M. Dalzell, S. McArdle, M. Berta, P. Bienias, C.-F. Chen, A. Gilyén, C. T. Hann, M. J. Kastoryano, E. T. Khabiboulline, A. Kubica, G. Salton, S. Wang, and F. G. S. L. Brandão, Quantum Algorithms: A Survey of Applications and End-to-End Complexities (Cambridge University Press, Cambridge, England, 2025), 10.1017/9781009639651.
12. A. Y. Kitaev, Quantum measurements and the Abelian stabilizer problem, arXiv:quant-ph/9511026.
13. K. M. Svore, M. B. Hastings, and M. Freedman, Faster phase estimation, Quantum Inf. Comput. 14, 306 (2014).
14. N. Wiebe and C. Granade, Efficient Bayesian phase estimation, Phys. Rev. Lett. 117, 010503 (2016).
15. L. Lin and Y. Tong, Heisenberg-limited ground-state energy estimation for early fault-tolerant quantum computers, PRX Quantum 3, 010318 (2022).
16. K. Wan, M. Berta, and E. T. Campbell, Randomized quantum algorithm for statistical phase estimation, Phys. Rev. Lett. 129, 030503 (2022).
17. L. Clinton, J. Bausch, J. Klassen, and T. Cubitt, Phase estimation of local Hamiltonians on NISQ hardware, New J. Phys. 25, 033027 (2023).
18. G. Wang, D. S. França, R. Zhang, S. Zhu, and P. D. Johnson, Quantum algorithm for ground state energy estimation using circuit depth with exponentially improved dependence on precision, Quantum 7, 1167 (2023).
19. J. Günther, F. Witteveen, A. Schmidhuber, M. Miller, M. Christandl, and A. Harrow, Phase estimation with partially randomized time evolution, arXiv:2503.05647.
20. S. Wang, S. McArdle, and M. Berta, Qubit-efficient randomized quantum algorithms for linear algebra, PRX Quantum 5, 020324 (2024).
21. P. K. Faehrmann, M. Steudtner, R. Kueng, M. Kieferova, and J. Eisert, Randomizing multi-product formulas for Hamiltonian simulation, Quantum 6, 806 (2022).
22. A. N. Chowdhury, G. H. Low, and N. Wiebe, A variational quantum algorithm for preparing quantum Gibbs states, arXiv:2002.00055.
23. Y. Wang, G. Li, and X. Wang, Variational quantum Gibbs state preparation with a truncated Taylor series, Phys. Rev. Appl. 16, 054035 (2021).
24. M. Consiglio, Variational quantum algorithms for Gibbs state preparation, in Numerical Computations: Theory and Algorithms, edited by Y. D. Sergeyev, D. E. Kvasov, and A. Astorino (Springer Nature Switzerland, Cham, 2025), pp. 56–70.
25. B. Bauer, D. Wecker, A. J. Millis, M. B. Hastings, and M. Troyer, Hybrid quantum-classical approach to correlated materials, Phys. Rev. X 6, 031045 (2016).
26. A. Baroni, J. Carlson, R. Gupta, A. C. Y. Li, G. N. Perdue, and A. Roggero, Nuclear two point correlation functions on a quantum computer, Phys. Rev. D 105, 074503 (2022).
27. M. L. Goh and B. Koczor, Direct estimation of the density of states for fermionic systems, arXiv:2407.03414.
28. Y. Subaş𝚤, L. Cincio, and P. J. Coles, Entanglement spectroscopy with a depth-two quantum circuit, J. Phys. A 52, 044001 (2019).
29. S. Subramanian and M.-H. Hsieh, Quantum algorithm for estimating α-Renyi entropies of quantum states, Phys. Rev. A 104, 022428 (2021).
30. S. P. Jordan, Fast quantum algorithms for approximating some irreducible representations of groups, arXiv:0811.0562.
31. D. Aharonov, V. Jones, and Z. Landau, A polynomial quantum algorithm for approximating the Jones polynomial, in Proceedings of the Thirty-Eighth Annual ACM Symposium on Theory of Computing, STOC ’06 (Association for Computing Machinery, New York, 2006), pp. 427–436.
32. P. W. Shor and S. P. Jordan, Estimating Jones polynomials is a complete problem for one clean qubit, Quantum Inf. Comput. 8, 681 (2008).
33. E. Knill and R. Laflamme, Power of one bit of quantum information, Phys. Rev. Lett. 81, 5672 (1998).
34. D. Wang, O. Higgott, and S. Brierley, Accelerated variational quantum eigensolver, Phys. Rev. Lett. 122, 140504 (2019).
35. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information: 10th Anniversary Edition (Cambridge University Press, Cambridge, England, 2010).
36. L. Clinton, T. S. Cubitt, R. Garcia-Patron, A. Montanaro, S. Stanisic, and M. Stroeks, Quantum phase estimation without controlled unitaries, arXiv:2410.21517.
37. S. Polla, G.-L. R. Anselmetti, and T. E. O’Brien, Optimizing the information extracted by a single qubit measurement, Phys. Rev. A 108, 012403 (2023).
38. T. E. O’Brien, S. Polla, N. C. Rubin, W. J. Huggins, S. McArdle, S. Boixo, J. R. McClean, and R. Babbush, Error mitigation via verified phase estimation, PRX Quantum 2, 020317 (2021).
39. Y. Zhou and Z. Liu, A hybrid framework for estimating nonlinear functions of quantum states, npj Quantum Inf. 10, 1 (2024).
40. H.-Y. Huang, R. Kueng, and J. Preskill, Predicting many properties of a quantum system from very few measurements, Nat. Phys. 16, 1050 (2020).
41. A. Elben, S. T. Flammia, H.-Y. Huang, R. Kueng, J. Preskill, B. Vermersch, and P. Zoller, The randomized measurement toolbox, Nat. Rev. Phys. 5, 9 (2023).
42. J. Morris and B. Dakić, Selective quantum state tomography, arXiv:1909.05880.
43. M. Paini, A. Kalev, D. Padilha, and B. Ruck, Estimating expectation values using approximate quantum states, Quantum 5, 413 (2021).
44. S. H. Sack, R. A. Medina, A. A. Michailidis, R. Kueng, and M. Serbyn, Avoiding barren plateaus using classical shadows, PRX Quantum 3, 020365 (2022).
45. H. H. S. Chan, R. Meister, M. L. Goh, and B. Koczor, Algorithmic shadow spectroscopy, PRX Quantum 6, 010352 (2025).
46. P. K. Faehrmann, Short-time simulation of quantum dynamics by Pauli measurements, Phys. Rev. A 112, 012602 (2025).
47. T. Schuster, J. Haferkamp, and H.-Y. Huang, Random unitaries in extremely low depth, Science 389, 92 (2025).
48. C. Bertoni, J. Haferkamp, M. Hinsche, M. Ioannou, J. Eisert, and H. Pashayan, Shallow shadows: Expectation estimation using low-depth random Clifford circuits, Phys. Rev. Lett. 133, 020602 (2024).
49. S. Chakraborty, Implementing any linear combination of unitaries on intermediate-term quantum computers, Quantum 8, 1496 (2024).
50. G. H. Low and I. L. Chuang, Hamiltonian simulation by qubitization, Quantum 3, 163 (2019).
51. G. H. Low and I. L. Chuang, Optimal Hamiltonian simulation by quantum signal processing, Phys. Rev. Lett. 118, 010501 (2017).
52. J. M. Martyn, Z. M. Rossi, A. K. Tan, and I. L. Chuang, Grand unification of quantum algorithms, PRX Quantum 2, 040203 (2021).
53. A. Gilyén, Y. Su, G. H. Low, and N. Wiebe, Quantum singular value transformation and beyond: Exponential improvements for quantum matrix arithmetics, in Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC 2019 (Association for Computing Machinery, New York, 2019), pp. 193–204.
54. See Supplemental Material at http://link.aps.org/supplemental/10.1103/cqjw-kl8s for additional information.
55. S. Chen, J. Cotler, H.-Y. Huang, and J. Li, Exponential separations between learning with and without quantum memory, in Proceedings of the 2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS) (IEEE, New York, 2022), pp. 574–585.
56. A. M. Childs and N. Wiebe, Hamiltonian simulation using linear combinations of unitary operations, Quantum Inf. Comput. 12, 901 (2012).
57. https://github.com/wilkensJ/quantum-circuit-drawio-library.